8.2 Applications to Geometry 1 Chapter 8. Eigenvalues: Further Applications and Computations 8.2. Applications to Geometry

Note. Figure 8.2 shows there intersections of planes with a double right-circular cone (“right-circular” because a plane perpendicular to the axis of the cone cuts the cone in a circle). The resulting “conic sections” shown are an ellipse (a circle is a special case of an ellipse), a hyperbola, and a parabola. In this section we state the standard equations for these three conic sections without any derivation. For an alternated approach where definitions are given in terms of sums and differences of certain distances (and the derivation of our formulae follow from these definitions) see my online Calculus 3 notes: http://faculty.etsu.edu/gardnerr/2110/notes -12e/c11s6.pdf. 8.2 Applications to Geometry 2

Note. The equation of an ellipse in standard form (that is, with center at (0, 0)) is x2 y2 + = 1. a2 b2 If the center of the ellipse is at the point (h, k) then the equation is

(x h)2 (y k)2 − + − = 1. a2 b2

See Figure 8.3.

2 2 Note. Every polynomial equation of the form c1x + c2y + c3x + c4y = d, where nonzero c1 an c2 are of the same sign, determines an ellipse. The center can be found by completing the square of the x2 and x terms and of the y2 and y terms. (x h)2 (y k)2 It’s possible for the equation to become of the form − + − = 0 in a2 b2 which case the solution set is the single point (h, k); this is called a degenerate (x h)2 (y k)2 ellipse. If the equation can be put in the form − + − = e < 0 then a2 b2 this is an empty ellipse. 8.2 Applications to Geometry 3

2 2 Note. Every polynomial equation of the form c1x + c2y + c3x + c4y = d, where nonzero c1 an c2 are of opposite signs, determines a hyperbola. The standard form of the equation of a hyperbola (with center (0, 0)) is

x2 y2 x2 y2 =1 or + = 1. a2 − b2 − a2 b2

These hyperbolas have asymptotes of y = (b/a)x; see Figure 8.4. If the center is ± not (0, 0) then completing the square (as with the ellipse) allows us to equations of the forms

(x h)2 (y k)2 (x h)2 (y k)2 − − =1 or − + − = 1. a2 − b2 − a2 b2

As with ellipses, a hyperbola may be degenerate (in which case the hyperbola consists of the straight lines y = (b/a)x). There is no empty hyperbola. ± 8.2 Applications to Geometry 4

Note. A polynomial equation of the form

2 2 c1x + c2x + c3y = d or c1y + c2x + c3y = d where c1 = 0, 6 determine a parabola. The standard form of the equation of a parabola (with vertex at (0, 0)) is ay = x2 or ax = y2 (see Figure 8.5). A parabola can can be degenerate (consisting only of the vertex, say when parameter a = 0 here) or empty (where parameter a< 0 here). Of course a parabola may have its vertex at a point (h, k) other than the origin.

2 2 Note. So, every equation of the form c1x + c2y + c3x + c4y = d with at least one of c1 or c2 nonzero describes a (possibly degenerate or empty) ellipse, hyperbola, or parabola. 8.2 Applications to Geometry 5

Note. We now turn our attention to equations of the form

ax2 + bxy + xy2 + dx + ey + f = 0 for a,b,c not all zero.

We use the results of the previous section to “transform away” the xy term, while still preserving the shape of the curve determined by the equation.

Theorem 8.2. Classification of Second-Degree Plane Curves. Every equation of the form

ax2 + bxy + xy2 + dx + ey + f = 0 for a,b,c not all zero can be reduced to an equation of the form

2 2 λ1t1 + λ2t2 + gt1 + ht2 + k = 0 by means of an orthogonal substitution corresponding to a rotation of the plane.

The coefficients λ1 and λ2 in the second equation are the eigenvalues of the sym- metric coefficient matrix of the quadratic-form portion of the first equation. The curve describes a (possibly degenerate or empty)

ellipse if λ1λ2 > 0

hyperbola if λ1λ2 < 0

parabola if λ1λ2 = 0.

Page 422 Example 8.2.2. Use rotation and translation of axes to sketch the curve 2xy + 2√2x = 1. Solution. To find the symmetric coefficient matrix for the cross term

u11 u12 u21 u22 2xy = 0x2 + 1xy + 1yx + 0y2 = x2 + xy + yx + y2, 2 2 2 2 8.2 Applications to Geometry 6 we take u11 = u22 = 0 and u12 = u21 = 2. Since aij = aji = uij/2 by Theorem

8.1.A, we have a11 a22 = 0 and a12 = a21 = 1 so that the symmetric coefficient − 111 112 0 1 matrix is A =   =  . For the eigenvalues of A consider  a21 a22   1 0 

0 λ 1 2 det(A λ )= − = λ 1 = 0 − I − 1 0 λ − and so the eigenvalues of A are λ1 = 1 and λ2 = 1. Since λ1λ2 = 1 < 0 then − − by Theorem 8.2, the equation determines a hyperbola. We need a basis for R2 consisting of normalized eigenvectors of A (Step 3 in “Diagonalizing a Quadratic

T Form f(~x)” of Section 8.1). For λ1 = 1 and ~v1 = [v1, v2] a corresponding − eigenvector, we need (A λ1 )~v1 = ~0, so consider − I

R2 R2 R1  1 1 0  → −  1 1 0  [a λ1 ~0] = ] − I |  1 1 0   0 0 0 

 1/√2  and so we take v1 = v2 and to normalize ~v1 we choose ~v1 = . Simil- −  1/√2  T − iarly, for λ2 = 1 and ~v2 = [v1, v2] a correpsonding eigenvector, we consider

R2 R2+R1 R1 R1  1 1 0  →  1 1 0  →−  1 1 0  [A λ1 ~0] = − ] − ] − − I | 1 1 0 0 0 0 0 0 0  −     

 1/√2  and so we take v1 = v2 and to normalize ~v2 we choose ~v2 = . So  1/√2  1/√2 1/√2 C =   (notice that det(C) = 1). As seen in the proof of the 1/√2 1/√2  −  8.2 Applications to Geometry 7

“Classification of Second-Degree Plane Curves,” Theorem 8.2, we take

x t1 1/√2 1/√2 t1 (1/√2)(t1 + t2)   = C   =     =   . y t2 1/√2 1/√2 t2 (1/√2)( t1 t2)      −     − − 

So with x = (1/√2)(t1 +t2) and y = (1/√2)( t1 +t2) we have that 2xy +2√2x = 1 − 1 1 1 2 2 becomes 2 (t1 +t2) ( t1 +t2)+2√2 (t1 +t2)=1or(t2 t1)+2(t1 +t2) = 1 √2 √2 − √2 − 2 2 2 2 or t1 + 2t1 +2 +2t2 =1or(t2 + 1) (t1 1) = 1. So this is in fact a hyperbola in − − −  1   1/√2   0  the (t1, t2)-coordinate system. Notice that C = and C = 0 1/√2 1    −     1/√2  so C rotates the x-axis 45◦ so that the t1-axis lies on the line y = x − −  1/√2  and C rotates the y-axis 45◦ so that the t2-axis lies on the line y = x. The − 2 2 center of hyperbola (t2 + 1) (t1 1) = 1 is then (1, 1) in the (t1, t2)-coordinate − − − system. The graph is as given in Figure 8.6. However, in the text, the eigenvalues are labeled λ1 = 1 and λ2 = 1 so that the t1- and t2-axes of Figure 8.6 do not − correspond to our t1- and t2-axis (rotate the t1- and t2-axes of Figure 8.6 by 90◦ − to get our axes). 8.2 Applications to Geometry 8

Definition. An equation in three variables of the form

2 2 2 c1x + c2y + c3z + c4z + c5y + c6z = d, where at least one of c1, c2, or c3 in nonzero, describes a quadric surface in 3-space (which might be degenerate or empty).

Note. Figures 8.7 through 8.15 (see the next page) show the eight possible quadric surfaces, along with equations. The equations are for quadric surfaces centered at the origin (0, 0, 0). By completing squares, we can convert these into quadric surfaces centered at point (h,k,m).

Note. Every equation of the form

ax2 + by2 + cz2 + dxy + exz + fyz + pz + qy + rz + s = 0 can be transformed into one of the forms of a quadric surface given in Figures 8.7 through 8.15. The technique is similar to that given in the proof of Theorem 8.2, “Classification of Second-Degree Plane Curves.” We summarize the claim in the following theorem. 8.2 Applications to Geometry 9 8.2 Applications to Geometry 10

Theorem 8.3. Principal Axis Theorem for R3. Every equation of the form

ax2 + by2 + cz2 + dxy + exz + fyz + pz + qy + rz + s = 0 can be reduced to an equation of the form

2 2 2 λ1t1 + λ2t2 + λ3t3 + p0t1 + q0t2 + r0t3 + s0 = 0 by an orthogonal substitution that corresponds to a rotation of axes.

Note. As in Theorem 8.2, we can classify which quadric surface is determined by

ax2 + by2 + cz2 + dxy + exz + fyz + pz + qy + rz + s = 0 in terms of the eigenvalues λ1, λ2, and λ3, we have the following (though some ambiguity remains):

Eigenvalues λ1, λ2, λ3 Quadric Surface

All of the same sign Ellipsoid Two of one sign and Elliptic cone, hyperboloid of one of the other sign two sheets, or hyperboloid of one sheet One zero, two of the same sign Elliptic paraboloid or elliptic cylinder (degenerate case) One zero, two of opposite signs Hyperbolic paraboloid or hyperbolic cylinder (degenerate case) Two zero, one nonzero Parabolic cylinder or two parallel planes (degenerate case)

Example. Page 430 Number 16. Revised: 6/23/2018